Force Exerted On A Small Body By A Small Body

When the distance R between the mass centers B* and B* of two (not necessarily rigid) bodies B and B exceeds in the case of each body the greatest distance from the mass center to_any point of the body, the system of gravitational forces exerted on B by B has a resultant F which can be expressed as

where a, is a unit vector directed from B* toward B*, G is the universal gravitational constant, and m and m are the masses of B and B, respectively, and where f° is a collection of terms of tth degree in |r|//?, f(0 is a collection of terms of ith degree in |r|/i?, and fm is a collection of terms in the product (|r|//?)'(|r|/i?)J, with r and r position vectors of generic points of B and B relative to B* and B*, respectively. In particular, f(2) = ^ {| [MI) - 5a, I • a,]a, + 31 • a,} (2)

and f<2) = [tr(I) - 5a, • I • a,]a, + 31 • a,} (3)

where I and I are the inertia dyadics of B for B* and of B for B*, respectively.

If a2 and a3 are defined so as to establish a dextral, orthogonal set of unit vectors a,, a2, a3, then f<2) and f<2) can be expressed in terms of these unit vectors and the moments and products of inertia of B and B for axes parallel to a,, a2, and a3 and passing through the mass centers of the individual bodies. To this end, IJk and IJk are defined as

after which f® and f,2) may be written f*2) = ¿[y Un + '33 - 2/,,)a, + /2,a2 + /3,a3] (6)

and f2) = V22 + /33 - 2/„)a, + /2ia2 + /3.a3j (7)

Alternatively, f(2) and f<2> can be expressed in terms of principal moments of inertia of B for B* and of B for B*. To accomplish this, two sets of dextral, orthogonal unit vectors, bi, b^, b3, and b,, b3, parallel to principal axes of inertia of B_ior B* and of B for B*, respectively, are introduced, and /y, Ij, Cjj, and Cij are defined as and

Thus one obtains

^ = -L (t M " 3C"2> + ^(1 - 3C.22) + /3(1 " 3C132)]a, mR 12

and f® = zr-r {4 [/.(I " 3C„2) + 72(1 - 3C.22) + 7,(1 - 3C,32)]a, mR2 12

A useful approximation to F in Eq. (1) may be obtained by defining F such that

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